hw05soln - 1 Homework 5(due November 15 Problem 5-1(3 pts...

1Homework 5 (due November 15)Problem 5-1 (3 pts.)Let the dispersion relationship for an ideal Bose gas particles be given by the following scaling law:ε(p)∼pαWhat should be the dimensionality of the physical spacedfor this system to be capable of Bose condensation?Solution:Bose condensation is possible ifµ= 0can be reached at afinite density, i.e. when the following integral converges:ρc=4π(2π~)3∞Z0∙expμ±(p)T¶−1¸−1p2dpDivergency is only possible at smallp.in this limit,[exp (±(p)/T)−1]∼pα.Hence, the following integral shouldconverge:PZ0pd−α−1dpWe conclude thatd > αProblem 5-2 (4 pts.)Find the energy densityUof a photonic gasinddimensions, as a function of temperatureT(assume all thefundamental constants to be the same as in our world). By using the standard thermodynamic relationships,find thecorresponding entropy densitys, free energy densityf, and radiation pressureP.

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